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The duration of an SIR epidemic on a configuration model
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We consider the spread of a stochastic SIR (Susceptible, Infec-tious, Recovered) epidemic on a configuration model random graph.We focus especially on the final stages of the outbreak and providelimit results for the duration of the entire epidemic, while we allowfor non-exponential distributions of the infectious period and for bothfinite and infinite variance of the asymptotic degree distribution in thegraph.

Our analysis relies on the analysis of some subcritical continuoustime branching processes and on ideas from first-passage percolation.

Keywords [en]
SIR epidemics, Time to extinction, Branching process approximation, First passage percolation
National Category
Mathematics
Research subject
Mathematical Statistics
Identifiers
URN: urn:nbn:se:su:diva-167360OAI: oai:DiVA.org:su-167360DiVA, id: diva2:1299549
Funder
Swedish Research CouncilAvailable from: 2019-03-27 Created: 2019-03-27 Last updated: 2019-03-29
In thesis
1. Stochastic epidemics on random networks
Open this publication in new window or tab >>Stochastic epidemics on random networks
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis considers stochastic epidemic models for the spread of epidemics in structured populations. The asymptotic behaviour of the models is analysed by using branching process approximations. The thesis contains four manuscripts.

Paper I is concerned with the study of the spread of sexually transmitted infections, or any other infectious diseases on a dynamic network. The model we investigate is about the spread of an SI (Susceptible → Infectious) type infectious disease in a population where partnerships are dynamic. We derive explicit formulas for the probability of extinction and the threshold parameter R0 using two branching process approximations for the model. In the first approximation some dependencies between infected individuals are ignored while the second branching process approximation is asymptotically exact and only defined if every individual in the population can have at most one partner at a time. By comparing the two approximations, we show that ignoring subtle dependencies in the dynamic epidemic model leads to wrong prediction of the probability of a large outbreak.

In paper II, we study a stochastic SIR (Susceptible → Infectious → Removed) epidemic model for the spread of an epidemic in populations structured through configuration model random graphs. We study the asymptotic (properly scaled) time until the end of an epidemic. This paper heavily relies on the theory of branching processes in continuous time.

In paper III, the effect of vaccination strategies on the duration of an epidemic in a large population is investigated. We consider three vaccination strategies: uniform vaccination, leaky vaccination and acquaintance vaccination.

In paper IV, we present a stochastic model for two successive SIR epidemics in the same network structured population. Individuals infected during the first epidemic might have (partial) immunity for the second one. The first epidemic is analysed through a bond percolation model, while the second epidemic is approximated by a three-type branching process in which the types of individuals depend on their status in the percolation clusters used for the analysis of the first epidemic. This branching process approximation enables us to calculate a threshold parameter and the probability of a large outbreak for the second epidemic. We use two special cases of acquired immunity for further evaluation.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2019
Keywords
Branching process, Configuration model, Random graph, Epidemic process, Final size, Threshold behaviour, Duration of an epidemic, Vaccination
National Category
Mathematics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-167373 (URN)978-91-7797-661-5 (ISBN)978-91-7797-662-2 (ISBN)
Public defence
2019-05-16, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 10:00 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 2: Manuscript. Paper 3: Manuscript. Paper 4: Manuscript.

Available from: 2019-04-23 Created: 2019-03-27 Last updated: 2019-04-09Bibliographically approved

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arXiv:1805.05117

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